Divide using synthetic division $$ ( -3x^{3}+2x-8 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&-3&0&2&-8\\& & 12& -48& \color{black}{184} \\ \hline &\color{blue}{-3}&\color{blue}{12}&\color{blue}{-46}&\color{orangered}{176} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+2x-8 }{ x+4 } = \color{blue}{-3x^{2}+12x-46} ~+~ \frac{ \color{red}{ 176 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-3&0&2&-8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ -3 }&0&2&-8\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-3&0&2&-8\\& & \color{blue}{12} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}-4&-3&\color{orangered}{ 0 }&2&-8\\& & \color{orangered}{12} & & \\ \hline &-3&\color{orangered}{12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 12 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-3&0&2&-8\\& & 12& \color{blue}{-48} & \\ \hline &-3&\color{blue}{12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -46 } $
$$ \begin{array}{c|rrrr}-4&-3&0&\color{orangered}{ 2 }&-8\\& & 12& \color{orangered}{-48} & \\ \hline &-3&12&\color{orangered}{-46}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -46 \right) } = \color{blue}{ 184 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-3&0&2&-8\\& & 12& -48& \color{blue}{184} \\ \hline &-3&12&\color{blue}{-46}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 184 } = \color{orangered}{ 176 } $
$$ \begin{array}{c|rrrr}-4&-3&0&2&\color{orangered}{ -8 }\\& & 12& -48& \color{orangered}{184} \\ \hline &\color{blue}{-3}&\color{blue}{12}&\color{blue}{-46}&\color{orangered}{176} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}+12x-46 } $ with a remainder of $ \color{red}{ 176 } $.